General Equation Of State Research Articles

Derivation of the General Equation of State for a Doped Semiconductor. The Equation is Solved Concerning the Electron Concentration in the Conduction Band for Different Concentrations and Properties of Doping Substances

The paper examines the state of a system consisting of a crystalline semiconductor doped with donor and acceptor impurities. The General Equation of State of the system is derived. Methods are proposed to solve the general equation for the concentration of electrons or holes in a semiconductor. The identity of the General Equations of State of semiconductors and aqueous solutions of electrolytes is discussed. An assumption is made about the practical application of the found relationships.

المحاكاة العددية للتدفقات المضغوطة المكو نة من عنصرين مع المعادلة العامة للحالة

Numerical simulations of compressible two-component flows with a general equation of state (EOS), written in the form of the Mie-Gru¨neisen EOS, have been conducted using the diffuse interface method on a structured mesh. The unsteady and inviscid one-dimensional sixequation model of Kapila is employed to describe compressible two-component flows. The model is hyperbolic and non-conservative. The solution of the hyperbolic equations, including the non-conservative equation for the volume fraction evolution, is obtained using an extended Harten-Lax-Leer (HLL) approximate Riemann solver. A general formulation for various EOSs is proposed to enable simulations of a wide range of applications. In which the two constituents are either governed by the same EOS or different types of EOS. In this model, both fluids have the same velocity, but each fluid has its own pressure. Therefore, the pressure relaxation process is performed instantaneously to drive both constituents' pressures towards equilibrium. Several computational test problems were conducted in one and two dimensions to show the ability of the numerical method to simulate such problems. The computed results are presented without introducing spurious pressure oscillations in the method.

Induced gravitational wave interpretation of PTA data: a complete study for general equation of state

We thoroughly study the induced gravitational wave interpretation of the possible gravitational wave background reported by PTA collaborations, considering the unknown equation of state w of the early universe. We perform a Bayesian analysis of the NANOGrav data using the publicly available PTArcade code together with SIGWfast for the numerical integration of the induced gravitational wave spectrum. We focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, we show that, while the results are not very sensitive to w when the GW peak is close to the PTA window, radiation domination is out of the 2σ contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the 2σ bounds yield 0.1 ≲ w ≲ 0.9 so that radiation domination is close to the central value. We also investigate the primordial black hole (PBH) abundance for both monochromatic and log-normal power spectrum. We show that, in general terms, a larger width and stiffer equation of state alleviates the overproduction of PBHs. No PBH overproduction requires w ≲ 0.57 up to 2-σ level for the monochromatic spectrum. Furthermore, including bounds from the cosmic microwave background, we find in general that the mass range of the PBH counterpart is bounded by 10-5 M ⊙ ≲ M PBH ≲ 10-1 M ⊙. Lastly, we find that the PTA signal can explain the microlensing events reported by OGLE for w ~ 0.7. Our work showcases a complete treatment of induced gravitational waves and primordial black holes for general w for future data analysis.

GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), meaning they consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by the BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system, which is the relativistic analogue of Godunov's symmetrizable form of the non-relativistic MHD system (Godunov, 1972 [19]). A key novelty in our schemes is the meticulous discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. Notably, we provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework (Wu and Shu, 2023 [49]), which equivalently converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. Finally, we implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios like ultra-relativistic blasts and jets in strongly magnetized environments.

Testing axionic dark matter during gravitational reheating

Assuming axions are potential dark matter (DM) candidate that make up all of the DM abundance, we discuss production of axions via (i) standard misalignment mechanism during the period of reheating and (ii) graviton-mediated 2-to-2 scattering of the inflaton and bath particles, where the inflaton ϕ oscillates in a monomial potential V(ϕ)∝ϕk with a general equation of state. Considering reheating takes place purely gravitationally, mediated by massless gravitons, we explore the viable region of the parameter space that agrees with the observed DM relic abundance, satisfying bounds from big bang nucleosynthesis and cosmic microwave background radiation. We also discuss complementarity between dedicated axion search experiments and futuristic gravitational wave search facilities in probing the viable parameter space, thanks to the presence of detectable primordial gravitational waves with an inflationary origin. Published by the American Physical Society 2024

Verification and validation of detonation-shock-dynamics relations for explosives described by general equation of state and chemical reaction models

Detonation shock dynamics is a powerful method to model the behaviour of High Explosives (HE). However in order to use this method, the underlying relationship between the local radius of curvature and the detonation speed must be known. Previous work has developed methods to calculate this effect using simple, single-step Arrhenius and polytropic gas, models for the chemical reaction and the equation of state, respectively. In recent years, more complex models for both reaction rates and equations of state have been developed which show better agreement with experimental data than these simple models, especially when considering condensed phase explosives.. This work presents the governing equations for solving these problems in a way that is generalised to use arbitrary equations of state as well as reaction models which may have more than a single step and multiple product species. This implementation is verified against exact solutions, demonstrating that the equations were implemented properly. The verified algorithm is then validated against experimental data and high fidelity simulations, showing that it is able to make accurate predictions in a regime where the underlying assumptions of the governing equations are valid. This approach has many applications: from creating equivalent detonation shock dynamics models for existing reactive burn calibrations for HE; to developing new functional forms and calibrations of reactive burn models for condensed phase high explosives.

A General Equation of State for a Quantum Simulator

A General Equation of State for a Quantum Simulator

High-Order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-Type Equation of State

High-Order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-Type Equation of State

Extension of an all-Mach Roe scheme able to deal with low Mach acoustics to full Euler system

We propose to extend the fix of Roe’s approximate Riemann solver developed for the Barotropic Euler equations in [2] to the full Euler equations. This scheme is built mainly to handle low Mach acousticwaves. Moreover, compared to pressure-centered type schemes, this numerical fix has the advantage of improving the numerical solution in the sense that the oscillating modes are reduced. The theoretical study is based on a two-time scales asymptotic analysis. It is proved that the Euler system equipped with a general equation of state is consistent with a first-order wave system in a low Mach number regime. Similar analysis is performed at the discrete level on the Roe scheme to derive the new fix. Numerical tests confirm the results obtained for the Barotropic case about the ability of this fix to deal with both steady and low Mach acoustic computations also in the case of full Euler equations.

Higgs inflation at the pole

We propose a novel possibility for Higgs inflation where the perturbative unitarity below the Planck scale is ensured by construction and the successful predictions for inflation are accommodated. The conformal gravity coupling for the Higgs field leads to the proximity of the effective Planck mass to zero in the Jordan frame during inflation, corresponding to a pole in the Higgs kinetic term in the Einstein frame. Requiring the Higgs potential to vanish at the conformal pole in the effective theory in the Jordan frame, we make a robust prediction of the successful Higgs inflation. For a successful Higgs inflation at the pole, we take the running quartic coupling for the Higgs field to be small enough at the inflation scale, being consistent with the low-energy data, but we need a nontrivial extension of the SM with extra scalar or gauge fields in order to keep the running Higgs quartic coupling small during inflation. Performing the perturbative analysis of reheating with the known couplings of the SM particles to the Higgs boson, we show that a concrete realization of the Higgs pole inflation can be pinned down by the reheating processes with a general equation of state for the Higgs inflaton. We illustrate some extensions of the simple Higgs pole inflation to the general pole expansions, the running Higgs quartic coupling in the Standard Model and its extension with a singlet scalar field, a supergravity embedding of the Higgs pole inflation.

Efficient solution of bimaterial Riemann problems for compressible multi-material flow simulations

When solving compressible multi-material flow problems, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular idea in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. The work presented in this paper aims to accelerate the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed, including (1) a change of integration variable through rarefaction fans, (2) storing and reusing integration trajectory data, (3) step size adaptation, and (4) constructing an R-tree on the fly to generate initial guesses. The performance of these acceleration methods is assessed using four example problems in underwater explosion, laser-induced cavitation, and hypervelocity impact. These problems exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the acceleration methods are able to significantly reduce the computational cost without affecting solver robustness or solution accuracy. In different cases, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times.

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Spatial eigenvalue problems for stars in hydrostatic equilibrium: Generalized Lane–Emden equations as boundary value problems

ABSTRACT We derive a generic spatial eigenvalue problem governing stars in hydrostatic equilibrium. Our approach generalizes the various Lane–Emden equations finding use over the past century, allowing for more general equations of state (EoS) while ensuring a stellar structure with finite size (without the need for artificial truncation of the radius). We show that the resulting stellar structure is encoded into two quantities: the eigenvalue, which determines the total size or mass of the star, and the density distribution, which encodes the internal structure. While our formalism recovers known results for polytrope and white dwarf EoS, we also study additional EoS, such as those incorporating excluded volumes or those calibrated through viral expansions. We obtain numerical values for the stellar structure under a variety of frameworks, comparing and contrasting stellar structure under different EoS. Interestingly, we show how different EoS can be calibrated to give solutions with the same stellar structure, highlighting the arbitrariness of a particular EoS for replicating observations. This leads us to comment on general properties EoS should obey to describe physically realistic stars. We also consider hydrostatic gas clouds immersed in larger regions having non-zero ambient density. We compare three analytical methods for finding solutions of these eigenvalue problems, including Taylor series solutions, the variational approximation, and the non-perturbative delta-expansion method. Although each method has benefits and drawbacks, we show that the delta-expansion method provides the most accuracy in replicating stellar structure.

Phase behavior of carbon dioxide + 2,4-dimethylpentane binary system at high pressures

The phase behavior of the branched alkane 2,4-dimethyl pentane (2,4-DMP) and carbon dioxide (CO2) binary mixture that has no published data, to the best of our knowledge, was studied. The liquid–vapor critical line was determined up to 121.4 bar and 422.35 K, as well as vapor–liquid equilibrium (VLE) data at different constant temperatures (323.15, 343.15, 363.15, and 383.15) K and pressures between (10.4 and 120.4) bar. Phase behavior investigation was carried out in a high-pressures (HP) cell fitted with a pair of sapphire windows, one acting as a piston, hence volume up to 60 cm3, utilizing a static-analytical approach with phases sampling by the so called ROLSI valves (rapid online sample injectors) connected to a gas chromatograph (GC) for composition determination. The isothermal and critical data points for the carbon dioxide + 2,4-dimethylpentane binary mixture were modeled with the cubic equations of state (EoSs), i.e., Soave–Redlich–Kwong (SRK), Peng–Robinson (PR), and General Equation of State (GEOS), in combination with van der Waals mixing rules (one- and two-parameter conventional mixing rules, 1PCMR and 2PCMR). The calculations demonstrated that the selected models are capable of correctly reproducing the phase behavior of the mixture under study.

New conserved integrals and invariants of radial compressible flow in n > 1 dimensions

Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid/gas flow in n > 1 dimensions. Apart from entropy, which is a well-known invariant, three additional invariants are found from an explicit determination of invariants up to first order. One holds for a general equation of state, and the two others hold only for entropic equations of state. A recursion operator on invariants is presented, which produces two hierarchies of higher-order invariants. Each invariant yields a corresponding integral invariant, describing an advected conserved integral on transported radial domains. In addition, a direct determination of kinematic conserved densities uncovers two ‘hidden’ non-advected conserved integrals: one describes enthalpy-flux, holding for barotropic equations of state; the other describes entropy-weighted energy, holding for entropic equations of state. A further explicit determination of a class of first-order conserved densities shows that the corresponding non-kinematic conserved integrals on transported radial domains are equivalent to integral invariants, modulo trivial densities.

Finding Singular Points of the Titration Curve by Solving the Equation of State of the Electrolyte Solution

Electrolyte solutions consisting of acids, bases, and their salts have one or more singular points. These special points include the equivalence points of the titration curves and the coordinates (composition) of the maximum buffer capacity of the buffer solution. Knowledge of these points is necessary for solving problems of analytical and preparative chemistry. This paper describes a method for finding singular points of an electrolyte solution by solving the general equation of state derived in our previous work.

A New Formulation of Generalized Equation of State (GEOS) based on Finite Strain Theory and Comparison with other Equations of State (EOSs)

Objective: To formulate the equation of state as well as their isothermal bulk modulus using the basic laws of thermodynamics and Identities. Method: This study has considered a relation of Eulerian finite strain having two arbitrary parameters a2 and b2 (a2,b2  1). The salient feature of this equation of state is that by substituting the values of parameters, the expressions for the prominent equations of state and their bulk modulus can be obtained. Findings : Four prototype solids viz. MgO, CaO, NaCl, and Al2O3 have been applied to the equation of state to test its validity and applicability. The results were compared with experimental data and other equations of state. Consequently, the proposed equation of state exhibits the same trend as prominent equations of state and provides better results. It corresponds well with the experimental curve at high pressure. Novelty: The GEOS can be used in the future for planning high-pressure experiments on the compression behavior of several materials, minerals, and solids. Keywords: The Eularian Finite Stain Theory; Equation of State; Prototype Solids; High Pressure

General Equation of State of Acid-Base Balance in Solutions: Resolution of the Equation with Respect to Hydrogen ion Concentrations and Concerning the Amount of Added Base or Acid. Titration Curves.

The paper considers the state of the system, which is an electrolyte solution containing an acid and a base in various ratios. The general equation of the state of the system is derived. Methods are proposed to solve the general equation for the concentration of hydrogen ions and for the base/acid introduced into the system. The theoretical dependencies are compared with experimental titration curves.

Existence of solutions to gas expansion problem through a sharp corner for 2‐D Euler equations with general equation of state

AbstractIn this article, we study the gas expansion problem by turning a sharp corner into a vacuum for the two‐dimensional (2‐D) pseudosteady compressible Euler equations with a convex equation of state. This problem can be considered as the interaction of a centered simple wave with a planar rarefaction wave. To obtain the global existence of a solution up to the vacuum boundary of the corresponding 2‐D Riemann problem, we consider several Goursat‐type boundary value problems for 2‐D self‐similar Euler equations and use the ideas of characteristic decomposition and bootstrap method. Further, we formulate 2‐D‐modified shallow water equations newly and solve a dam‐break‐type problem for them as an application of this work. Moreover, we also recover the results from the available literature for certain equations of states that provide a check that the results obtained in this article are actually correct.

Ultraviolet freeze-in dark matter through the dilaton portal

We study a class of models in which the Standard Model (SM) and dark matter (DM) belong to a conformal/scale-invariant theory at high energies. Scale invariance is spontaneously broken at scale $f$, giving rise to a dilaton as the corresponding Goldstone boson. In the low-energy theory, we assume that DM interacts with the SM solely through the dilaton portal, which is suppressed by the conformal breaking scale $f$. For $f\ensuremath{\gg}\mathrm{TeV}$, the portal interactions are extremely weak, resulting in DM not being in thermal equilibrium with the SM. Thus, ultraviolet freeze-in production of DM occurs through the dilaton portal, being most effective at the maximum temperature of the SM bath. The temperature evolution is greatly impacted by the reheating dynamics, which we parametrize using a general equation of state $w$ and temperature at the end of reheating ${T}_{\mathrm{rh}}$. We analyze the implications of the reheating dynamics for DM production in this framework and identify regions of parameter space that result in the observed DM relic abundance for a wide range of DM masses and reheating temperatures for scalar, vector, or fermion DM.